Young Whan Lee

On the stability of a quadratic Jensen type functional equation

Abstract In this paper we obtain the general solution of the quadratic Jensen type functional equation 9f x+y+z 3 +f(x)+f(y)+f(z)=4 f x+y 2 +f y+z 2 +f z+x 2 and prove the stability of this equation in the spirit of Hyers, Ulam, Rassias, and Găvruta.

Boundedness of approximate trigonometric functions

Abstract In this work, we prove that approximate trigonometric functions are bounded. That is, if a non-zero function f satisfies the inequality | f ( x + y ) − f ( x − y ) − 2 f ( x ) f ( y ) | ≤ φ ( x ) or φ ( y ) , then f is bounded.

Solution and Hyers-Ulam-Rassias Stability of Generalized Mixed Type Additive-Quadratic Functional Equations in Fuzzy Banach Spaces

By using fixed point methods and direct method, we establish the generalized Hyers-Ulam stability of the following additive-quadratic functional equation for fixed integers with in fuzzy Banach spaces.

Superstability and Stability of the Pexiderized Multiplicative Functional Equation

We obtain the superstability of the Pexiderized multiplicative functional equation and investigate the stability of this equation in the following form: .

Superstability of the functional equation with a cocycle related to distance measures

In this paper, we obtain the superstability of the functional equation f(pr,qs)+f(ps,qr)=θ(pq,rs)f(p,q)f(r,s) for all p,q,r,s∈G, where G is an Abelian group, f a functional on G2, and θ a cocycle on G2. This functional equation is a generalized form of the functional equation f(pr,qs)+f(ps,qr)=f(p,q)f(r,s), which arises in the characterization of symmetrically compositive sum-form distance measures, and as products of some multiplicative functions. In reduction, they can be represented as exponential functional equations. Also we investigate the superstability with following functional equations: f(pr,qs)+f(ps,qr)=θ(pq,rs)f(p,q)g(r,s), f(pr,qs)+f(ps,qr)=θ(pq,rs)g(p,q)f(r,s), f(pr,qs)+f(ps,qr)=θ(pq,rs)g(p,q)g(r,s), f(pr,qs)+f(ps,qr)=θ(pq,rs)g(p,q)h(r,s).MSC:39B82, 39B52.

Approximate pexiderized gamma-beta type functions

AbstractWe show that every unbounded approximate pexiderized gamma-beta type function has a gamma-beta type. That is, we obtain the superstability of the pexiderized gamma-beta type functional equation β(x,y)f(x+y)=g(x)h(y) and also investigate the superstability as the following form: |β(x,y)f(x+y)g(x)h(y)−1|≤φ(x,y).MSC:39B72, 38B22, 39B82.

Fixed points and stability of functional equations in fuzzy ternary Banach algebras

AbstractBy using Diaz and Margolis fixed point theorem, we establish the generalized Hyers-Ulam-Rassias stability of the ternary homomorphisms and ternary derivations between fuzzy ternary Banach algebras associated to the following (m,n)-Cauchy-Jensen additive functional equation: ∑1≤i1<⋯<im≤n1≤kl≤nkl≠ij,∀j∈{1,…,m}f(∑j=1mxijm+∑l=1n−mxkl)=(n−m+1)n(nm)∑i=1nf(xi).MSC:39B52, 46S40, 26E50.

Approximation of Mixed-Type Functional Equations in Menger PN-Spaces

Let and be vector spaces. We show that a function with satisfies for all , if and only if there exist functions , and such that for all , where the function is symmetric for each fixed one variable and is additive for fixed two variables, is symmetric bi-additive, is additive and (, ) for all . Furthermore, we solve the stability problem for a given function satisfying , in the Menger probabilistic normed spaces.

SUPERSTABILITY OF A GENERALIZED EXPONENTIAL FUNCTIONAL EQUATION OF PEXIDER TYPE

We obtain the superstability of a generalized exponential functional equation f(x + y) = E(x, y)g(x)f(y) and investigate the stability in the sense of R. Ger [4] of this equation in the following setting: ∣∣∣∣ f(x + y) E(x, y)g(x)f(y) − 1 ∣∣∣∣ ≤ φ(x, y), where E(x, y) is a pseudo exponential function. From these results, we have superstabilities of exponential functional equation and Cauchy’s gamma-beta functional equation.

CONTINUITY OF AN APPROXIMATE JORDAN MAPPING

We show that every Jordan functional on a Banach algebra A is continuous. From this result we obtain that every Jordan mapping from A into a continuous function space C(S) is continuous and its norm less than or equal where S is a compact Hausdorff space. This is a generalization of Jaroszs result [3, Proposition 5.5].